We construct a solution to a 2 × 2 strictly hyperbolic system of conservation laws, showing that the Godunov scheme  can produce an arbitrarily large amount of oscillations. This happens when the speed of a shock is close to rational, inducing a resonance with the grid. Differently from the Glimm scheme or the vanishing-viscosity method, for systems of conservation laws our counterexample indicates that no a priori BV bounds or L 1-stability estimates can in general be valid for finite difference schemes.
All Science Journal Classification (ASJC) codes
- Applied Mathematics